Images Of Some ε-derivations And Mathieu-zhao Subspaces

Let K be an algebracally closed field of characteristic zero and K[x]is the polynomial algebra over K,where x=(x1,x2,…,xn)is a commuting variable,δ denote a nonzero ε-derivation of K[x],δ=I-φ.In this paper we prove that the LFED Conjecture is ture for polynomial algebras in two variables with ε-derivation δ,φ is a linear polynomial homomorphism or affine polynomial homomorphism.For polynomial algebras in three or four variables,we show that the LFED Conjecture is ture for someε-derivations δ and extend the conclusion to the higher dimensional case.